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Chapter 11: Problem 56

Replace the Cartesian equations in Exercises \(53-66\) with equivalent polar equations. $$x-y=3$$

Short Answer

Expert verified
The polar equation is \(r(\cos \theta - \sin \theta) = 3\).

Step by step solution

01

Understand the Relationship between Cartesian and Polar Coordinates

Cartesian coordinates are defined as \((x, y)\), where \(x\) and \(y\) are the horizontal and vertical distances from the origin, respectively. Polar coordinates are defined as \((r, \theta)\), where \(r\) is the distance from the origin to the point, and \(\theta\) is the angle from the positive x-axis to the line connecting the origin to the point. The relationships are given by the formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\).
02

Substitute Polar Coordinates into the Cartesian Equation

Take the given Cartesian equation \(x - y = 3\) and substitute the polar formulas for \(x\) and \(y\): \(r \cos \theta - r \sin \theta = 3\).
03

Simplify the Polar Equation

Factor out \(r\) from the left side of the equation: \(r(\cos \theta - \sin \theta) = 3\). This is the equivalent polar equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cartesian to Polar Conversion
Converting between Cartesian and polar coordinates is a fundamental skill in various fields of mathematics and physics. Typically in Cartesian coordinates, we describe a point in a two-dimensional plane using
  • two values:
    • its horizontal offset from the origin, noted as the x-coordinate, and
    • its vertical offset, which is the y-coordinate.
    This system is intuitive and useful for many practical applications.
In contrast, polar coordinates describe a point by specifying
  • the distance from the origin, represented by \( r \), and
  • the angle from the positive x-axis, denoted as \( \theta \).
To switch between these systems, you apply the equations:
  • \( x = r \cos \theta \)
  • \( y = r \sin \theta \)
By inserting these polar relationships into a Cartesian equation, you transform it into an equivalent polar form.
Coordinate Systems
Understanding different coordinate systems is vital for graphing and analyzing mathematical equations. The two primary systems are Cartesian and polar.
  • **Cartesian System**: This uses perpendicular axes (x and y) to create a grid. Points are identified by their distances from these axes.
  • **Polar System**: This system uses a point and an angle. The point's position is determined by how far it is from the origin and its angle relative to the x-axis.
Each system offers unique benefits. The Cartesian system is straightforward when dealing with straight lines and rectangular shapes.
On the other hand, the Polar system excels at describing curves and circles, as it naturally incorporates rotational symmetry.
Equations in Polar Form
Equations written in polar form make it easier to analyze patterns and symmetries of curves. A polar equation relates the radius, \( r \), to the angle \( \theta \), providing a different perspective compared to Cartesian equations.
  • **Simplicity**: Polar equations often simplify complex curves. For example, circles and spirals have neat equations in polar form.
  • **Applications**: They're useful in fields like engineering and physics where rotational dynamics are common tasks.
To convert from Cartesian to polar form, replace the Cartesian equations with polar relations, such as in the example equation: Given \( x - y = 3 \), converting it to polar form gives \( r (\cos \theta - \sin \theta) = 3 \).
This translation helps in visualizing how lines and curves behave in the polar coordinate system.

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Most popular questions from this chapter

Find the areas of the surfaces generated by revolving the curves about the indicated axes. $$ x=t+\sqrt{2}, \quad y=\left(t^{2} / 2\right)+\sqrt{2} t,-\sqrt{2} \leq t \leq \sqrt{2} ; \quad y-\text {axis} $$

Find polar equations for the circles in Exercises \(57-64 .\) Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations. $$(x+2)^{2}+y^{2}=4$$

Exercises \(45-48\) give equations for parabolas and tell how many units up or down and to the right or left each parabola is to be shifted. Find an equation for the new parabola, and find the new vertex, focus, and directrix. $$ x^{2}=8 y, \quad \text { right } 1, \text { down } 7 $$

The witch of Maria Agnesi The bell-shaped witch of Maria Agnesi can be constructed in the following way. Start with a circle of radius \(1,\) centered at the point \((0,1),\) as shown in the accompanying figure. Choose a point \(A\) on the line \(y=2\) and connect it to the origin with a line segment. Call the point where the segment crosses the circle \(B\) . Let \(P\) be the point where the vertical line through \(A\) crosses the horizontal line through \(B\) . The witch is the curve traced by \(P\) as \(A\) moves along the line \(y=2\) . Find parametric equations and a parameter interval for the witch by expressing the coordinates of \(P\) in terms of \(t\) the radian measure of the angle that segment \(O A\) makes with the positive \(x\) -axis. The following equalities (which you may assume) will help. $$\begin{array}{ll}{\text { a. } x=A Q} & {\text { b. } y=2-A B \sin t} \\\ {\text { c. } A B \cdot O A=(A Q)^{2}}\end{array}$$

Exercises \(35-38\) give information about the foci, vertices, and asymptotes of hyperbolas centered at the origin of the \(x y\) -plane. In each case, find the hyperbola's standard-form equation from the information given. $$ \begin{array}{l}{\text { Vertices: }(0, \pm 2)} \\ {\text { Asymptotes: } y=\pm \frac{1}{2} x}\end{array} $$
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